January  2012, 32(1): 265-291. doi: 10.3934/dcds.2012.32.265

Non-autonomous 3D primitive equations with oscillating external force and its global attractor

1. 

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  July 2010 Revised  December 2010 Published  September 2011

In this article, we consider a non-autonomous three-dimensional primitive model of the ocean with a singularly oscillating external force depending on a small parameter $ \epsilon. $ We prove the existence of the uniform global attractor $\mathcal{A}^{\epsilon} $ in $V,$ (i.e., with the $H^1-$regularity). Furthermore, using the method of [13] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $\mathcal{A}^{\epsilon} $ as $ \epsilon $ goes to zero.
Citation: T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co., (1992). Google Scholar

[2]

C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension,, Adv. Math. Sci. Appl., 4 (1994), 465. Google Scholar

[3]

C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model,, Comm. Pure Appl. Math., 56 (2003), 198. doi: 10.1002/cpa.10056. Google Scholar

[4]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math. (2), 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar

[5]

T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17. doi: 10.3934/dcdss.2009.2.17. Google Scholar

[6]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. Google Scholar

[7]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating coefficients, International Conference on Differential and Functional Differential Equations (Moscow, 1999),, Funct. Differ. Equ., 8 (2001), 123. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math., 192 (2001), 11. doi: 10.1070/SM2001v192n01ABEH000534. Google Scholar

[10]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces,, J. Math. Pures Appl. (9), 90 (2008), 469. doi: 10.1016/j.matpur.2008.07.001. Google Scholar

[11]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces,, Nonlinearity, 22 (2009), 351. doi: 10.1088/0951-7715/22/2/006. Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002). Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor,, J. Dynam. Differential Equations, 19 (2007), 655. Google Scholar

[14]

V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging,, Discrete Contin. Dyn. Syst., 12 (2005), 27. Google Scholar

[15]

A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55. doi: 10.3934/dcdss.2009.2.55. Google Scholar

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. Google Scholar

[17]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology,", John Wiley and Sons, (1980). Google Scholar

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A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17,, Mason, (1991). Google Scholar

[19]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption,, Nonlinear Anal., 61 (2005), 425. doi: 10.1016/j.na.2004.12.005. Google Scholar

[20]

C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations,, Chin. Ann. of Math. Ser. B, 23 (2002), 277. doi: 10.1142/S0252959902000262. Google Scholar

[21]

C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis,, Discrete Contin. Dyn. Syst., 9 (2003), 97. Google Scholar

[22]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar

[23]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system,, J. Differential Equations, 240 (2007), 249. Google Scholar

[24]

P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: 10.1023/A:1019156812251. Google Scholar

[25]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dyn. Continuous Impulsive Systems, 4 (1998), 211. Google Scholar

[26]

G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283. Google Scholar

[27]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar

[28]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. doi: 10.1088/0951-7715/5/5/002. Google Scholar

[29]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I),, Computational Mechanics Advance, 1 (1993), 3. Google Scholar

[30]

J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII),, Computational Mechanics Advance, 1 (1993), 55. Google Scholar

[31]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII),, Math. Pures et Appl., 73 (1995), 105. Google Scholar

[32]

J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday,, Nonlinear Anal. Ser. A. Theory Methods, 40 (2000), 439. doi: 10.1016/S0362-546X(00)85026-9. Google Scholar

[33]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. Google Scholar

[34]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. Google Scholar

[35]

T. Tachim Medjo, On the uniqueness of $z$-weak solutions of the three-dimensional primitive equations of the ocean,, Nonlinear Anal. Real World Appl., 11 (2010), 1413. doi: 10.1016/j.nonrwa.2009.02.031. Google Scholar

[36]

J. Pedlosky, "Geophysical Fluid Dynamics,", Second edition, (1987). Google Scholar

[37]

J. P. Peixoto and A. H. Oort, "Physics of Climate,", American Institute of Physics, (1992). Google Scholar

[38]

R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation,, Appl. Anal., 70 (1998), 147. doi: 10.1080/00036819808840682. Google Scholar

[39]

E. Simmonet, T. Tachim Medjo and R. Temam, Barotropic-baroclinic formulation of the primitive equations of the ocean,, Applicable Analysis, 82 (2003), 439. doi: 10.1080/0003681031000094591. Google Scholar

[40]

H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations,, Nonlinearity, 22 (2009), 667. doi: 10.1088/0951-7715/22/3/008. Google Scholar

[41]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Vol. 68, Second edition,, Appl. Math. Sci., (1988). Google Scholar

[42]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar

[43]

S. Wang, "On Solvability for the Equations of the Large-Scale Atmospheric Motion,", Ph.D thesis, (1988). Google Scholar

[44]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations,, Dyn. Syst., 23 (2008), 1. doi: 10.1080/14689360701611821. Google Scholar

[45]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling,", Oxford University Press, (1986). Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25,, North-Holland Publishing Co., (1992). Google Scholar

[2]

C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier-Stokes equations and estimate of its dimension,, Adv. Math. Sci. Appl., 4 (1994), 465. Google Scholar

[3]

C. Cao and E. S. Titi, Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model,, Comm. Pure Appl. Math., 56 (2003), 198. doi: 10.1002/cpa.10056. Google Scholar

[4]

C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,, Ann. of Math. (2), 166 (2007), 245. doi: 10.4007/annals.2007.166.245. Google Scholar

[5]

T. Caraballo and P. E. Kloeden, Non-autonomous attractor for integro-differential evolution equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 17. doi: 10.3934/dcdss.2009.2.17. Google Scholar

[6]

T. Caraballo and J. Real, Asymptotic behavior of two-dimensional Navier-Stokes equations with delays,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3181. doi: 10.1098/rspa.2003.1166. Google Scholar

[7]

T. Caraballo and J. Real, Attractors for 2D Navier-Stokes models with delays,, J. Differential Equations, 205 (2004), 271. Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating coefficients, International Conference on Differential and Functional Differential Equations (Moscow, 1999),, Funct. Differ. Equ., 8 (2001), 123. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Averaging of trajectory attractors of evolution equations with rapidly oscillating terms,, Sb. Math., 192 (2001), 11. doi: 10.1070/SM2001v192n01ABEH000534. Google Scholar

[10]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of nonautonomous damped wave equations with singularly oscillating external forces,, J. Math. Pures Appl. (9), 90 (2008), 469. doi: 10.1016/j.matpur.2008.07.001. Google Scholar

[11]

V. V. Chepyzhov, V. Pata and M. I. Vishik, Averaging of 2D Navier-Stokes equations with singularly oscillating forces,, Nonlinearity, 22 (2009), 351. doi: 10.1088/0951-7715/22/2/006. Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49,, American Mathematical Society, (2002). Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Non-autonomous 2D Navier-Stokes system with singularly oscillating external force and its global attractor,, J. Dynam. Differential Equations, 19 (2007), 655. Google Scholar

[14]

V. V. Chepyzhov, M. I. Vishik and W. L. Wendland, On non-autonomous Sine-Gordon type equations with a simple global attractor and some averaging,, Discrete Contin. Dyn. Syst., 12 (2005), 27. Google Scholar

[15]

A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55. doi: 10.3934/dcdss.2009.2.55. Google Scholar

[16]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 2 (1995), 307. Google Scholar

[17]

G. J. Haltiner and R. T. Williams, "Numerical Prediction and Dynamic Meteorology,", John Wiley and Sons, (1980). Google Scholar

[18]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Recherches en Mathématiques Appliquées, 17,, Mason, (1991). Google Scholar

[19]

C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption,, Nonlinear Anal., 61 (2005), 425. doi: 10.1016/j.na.2004.12.005. Google Scholar

[20]

C. Hu, R. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations,, Chin. Ann. of Math. Ser. B, 23 (2002), 277. doi: 10.1142/S0252959902000262. Google Scholar

[21]

C. Hu, R. Temam and M. Ziane, The primitive equations of the large scale ocean under the small depth hypothesis,, Discrete Contin. Dyn. Syst., 9 (2003), 97. Google Scholar

[22]

N. Ju, The global attractor for the solutions to the 3D viscous primitive equations,, Discrete Contin. Dyn. Syst., 17 (2007), 159. doi: 10.3934/dcds.2007.17.159. Google Scholar

[23]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system,, J. Differential Equations, 240 (2007), 249. Google Scholar

[24]

P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Numer. Algorithms, 14 (1997), 141. doi: 10.1023/A:1019156812251. Google Scholar

[25]

P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dyn. Continuous Impulsive Systems, 4 (1998), 211. Google Scholar

[26]

G. M. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations,, C. R. Math. Acad. Sci. Paris, 343 (2006), 283. Google Scholar

[27]

J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications,, Nonlinearity, 5 (1992), 237. doi: 10.1088/0951-7715/5/2/001. Google Scholar

[28]

J. L. Lions, R. Temam and S. Wang, On the equations of large-scale ocean,, Nonlinearity, 5 (1992), 1007. doi: 10.1088/0951-7715/5/5/002. Google Scholar

[29]

J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I),, Computational Mechanics Advance, 1 (1993), 3. Google Scholar

[30]

J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII),, Computational Mechanics Advance, 1 (1993), 55. Google Scholar

[31]

J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAOIII),, Math. Pures et Appl., 73 (1995), 105. Google Scholar

[32]

J. L. Lions, R. Temam and S. Wang, On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case. Lakshmikantham's legacy: A tribute on his 75th birthday,, Nonlinear Anal. Ser. A. Theory Methods, 40 (2000), 439. doi: 10.1016/S0362-546X(00)85026-9. Google Scholar

[33]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. Google Scholar

[34]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. Google Scholar

[35]

T. Tachim Medjo, On the uniqueness of $z$-weak solutions of the three-dimensional primitive equations of the ocean,, Nonlinear Anal. Real World Appl., 11 (2010), 1413. doi: 10.1016/j.nonrwa.2009.02.031. Google Scholar

[36]

J. Pedlosky, "Geophysical Fluid Dynamics,", Second edition, (1987). Google Scholar

[37]

J. P. Peixoto and A. H. Oort, "Physics of Climate,", American Institute of Physics, (1992). Google Scholar

[38]

R. Samelson, R. Temam and S. Wang, Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation,, Appl. Anal., 70 (1998), 147. doi: 10.1080/00036819808840682. Google Scholar

[39]

E. Simmonet, T. Tachim Medjo and R. Temam, Barotropic-baroclinic formulation of the primitive equations of the ocean,, Applicable Analysis, 82 (2003), 439. doi: 10.1080/0003681031000094591. Google Scholar

[40]

H. Song, S. Ma and C. Zhong, Attractors of non-autonomous reaction-diffusion equations,, Nonlinearity, 22 (2009), 667. doi: 10.1088/0951-7715/22/3/008. Google Scholar

[41]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Vol. 68, Second edition,, Appl. Math. Sci., (1988). Google Scholar

[42]

R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", AMS-Chelsea Series, (2001). Google Scholar

[43]

S. Wang, "On Solvability for the Equations of the Large-Scale Atmospheric Motion,", Ph.D thesis, (1988). Google Scholar

[44]

Y. Wang and C. Zhong, On the existence of pullback attractors for non-autonomous reaction-diffusion equations,, Dyn. Syst., 23 (2008), 1. doi: 10.1080/14689360701611821. Google Scholar

[45]

W. M. Washington and C. L. Parkinson, "An Introduction to Three-Dimensional Climate Modeling,", Oxford University Press, (1986). Google Scholar

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